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This document contains a post-print version of the paper

Modeling and static optimization of a variable speed pumped storagepower plant

authored by J. Schmidt, W. Kemmetmüller, and A. Kugi

and published in Renewable Energy.

The content of this post-print version is identical to the published paper but without the publisher’s final layout orcopy editing. Please, scroll down for the article.

Cite this article as:J. Schmidt, W. Kemmetmüller, and A. Kugi, “Modeling and static optimization of a variable speed pumped storagepower plant”, Renewable Energy, vol. 111, pp. 38–51, 2017. doi: https://dx.doi.org/10.1016/j.renene.2017.03.055

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@Article{Schmidt_2017_Renewable_Energy,author = {Schmidt, J. and Kemmetm\"uller, W. and Kugi, A.},title = {{M}odeling and static optimization of a variable speed pumped storage power plant},journal = {Renewable Energy},year = {2017},volume = {111},pages = {38-51},doi = {https://dx.doi.org/10.1016/j.renene.2017.03.055},

}

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Modeling and Static Optimization of a Variable Speed Pumped Storage Power Plant

J. Schmidt∗, W. Kemmetmuller∗, A. Kugi∗

Automation and Control Institute, Vienna University of Technology, Gußhausstraße 27–29, 1040 Vienna, Austria

Abstract

Pumped storage power plants are key components to stabilize electric distribution networks with high amount of in-termittent power sources as, e.g., solar and wind power plants. Tailored mathematical models are important for thetransient and the stationary analysis of such plants. A comprehensive mathematical model of a variable speed operatedpumped storage power plant, which incorporates reversible pump turbines in combination with doubly fed inductionmachines, is developed in this paper. Special emphasis is laid on an accurate description of important dynamic effects(e.g., water hammer) and of the energy losses of the system. Based on this model, optimal stationary operating pointsare determined, which minimize the overall system losses and systematically take into account the operating constraints.

Keywords: variable speed pumped storage power plant, doubly fed induction machine, mathematical modeling,optimal stationary operation

1. Introduction

Electric power industry is currently faced with environ-mental issues, increasing energy consumption and limitedresources of fossil fuels. These challenges are, amongst ot-hers, tackled by an intensified usage of renewable energysources, in particular wind and sun. The intermittent na-ture of these energy sources calls for a sufficient amount oflarge scale energy storage capabilities in order to ensure ageneration-load balance and thus grid stability. The well-established pumped storage power plants (PSPPs) stillrepresent the most attractive way of large scale energystorage, having a worldwide installed capacity of approxi-mately 130 GW [1]. In particular variable speed operatedPSPPs with reversible pump turbines offer distinct advan-tages in comparison to conventional fixed-speed PSPPs,including: (i) increased efficiency (especially during part-load operation) and an enhanced operating range in tur-bine mode, (ii) improved network frequency regulation ca-pabilities due to rapid injection of active power (flywheeleffect) as well as (iii) improved active power regulationcapability during pump operation, see, e.g., [2, 3, 4, 5].

Together with the ability of reactive power control, thesefeatures make variable speed PSPPs an excellent aid forimproving grid stability, e.g., by primary frequency con-trol. Two types of variable speed PSPPs are typicallyconsidered in literature: the doubly fed induction machine(DFIM) with a part-load converter at the rotor terminals

∗Corresponding author. Tel.: +43 1 58801 376296, Fax: +43 158801 9376296.

Email addresses: [email protected] (J. Schmidt),[email protected] (W. Kemmetmuller),[email protected] (A. Kugi)

and the synchronous machine with a full-load converterat the stator terminals [5, 6, 7]. The converter fed syn-chronous machine inherently offers some advantages overthe DFIM, including a much larger possible speed range[7]. While power electronics components typically limitthe power range of a full-load converter, the recent ad-vances in power electronics technology allow to build suchPSPPs with ever increasing power, see, e.g., [8] where acommissioned PSPP applying a 100 MW full-load conver-ter is described.

Nevertheless, the usual restriction of the speed range toapproximately ±10 % around the synchronous speed al-lows to size the part-load converter of the DFIM topologyto only a small portion of the rated machine power. Thisconstitutes a decisive economic advantage of the DFIMtopology, currently making it the predominant technologyfor high power (& 100 MW) applications, with several ex-amples of commissioned power plants, see, e.g., [9, 10, 11].Hence it is a variable speed PSPP employing the DFIMtopology that is considered in the present contribution.

Mathematical models of the PSPP are required for thedynamic simulation, the controller design and the optimi-zation of the operation of PSPPs. Thus, one goal of thiscontribution is the derivation of a comprehensive mathe-matical model of a variable speed PSPP. In particular,special focus is laid on the accurate description of the dy-namic behavior as well as of the losses of the overall plant,incorporating both the mechanic as well as the electric keycomponents. In literature, see, e.g., [10, 12, 13], simplifiedpump turbine models are often used, which are not suit-able to properly reflect pump turbine losses in a largeroperating range. A more sophisticated PSPP model in-tended for power system simulations is presented in [14].

Preprint submitted to Renewable Energy April 12, 2017

Post-print version of the article: J. Schmidt, W. Kemmetmüller, and A. Kugi, “Modeling and static optimization of a variable speedpumped storage power plant”, Renewable Energy, vol. 111, pp. 38–51, 2017. doi: https://dx.doi.org/10.1016/j.renene.2017.03.055The content of this post-print version is identical to the published paper but without the publisher’s final layout or copy editing.


C

DFIM

DFIMgrid

GSIiCdq1

iCdq2

iDdq1KiDdq1

iDdq2

iSdq1 iSdq2

lowerreservoir

plant unitsA and B

Pgrid , Qgrid

S

upperreservoir

uCdq1

uCdq2

uDdq1KuD

dq1

uDdq2

uSdq1 uS

dq2

uDC RSI

VSC1 2

3

4

5

6

78

9

10

11 12

Figure 1: Setup of the considered variable speed pumped storage power plant (PSPP). The pipe segment numbering corresponds to Table 3.The electric system of each plant consists of a DFIM, a step-up transformer (S), a converter transformer (C) and a voltage-source converter(VSC), comprising a grid side inverter (GSI) and a rotor side inverter (RSI). Vector notation is applied for the respective dq-quantities, e.g.,

uDdq1 =

[uDd1 uDq1

]T.

Here, the hydraulic system is modeled in a similar wayas in this paper, i.e. modeling the pump turbine in aquasi-stationary manner by its characteristics and nume-rically solving the pipe model’s partial differential equa-tions (PDEs) by the Method of Characteristics (MOC).The mathematical model of the electric system, however,is not described in detail. This might be reasonable forpower system simulations, but is not appropriate for con-troller design and optimization. In [4] a variable speedPSPP is modeled based on the hydraulic modeling appro-ach of [15, 16], which applies a finite difference scheme forthe numerical solution of the pipe PDEs. The simulationbuilds upon an existing software tool, which was originallydeveloped for the simulation of electric systems [16]. Thus,also the hydraulic components are described by equiva-lent electric models. As the pump turbine characteristicsare interpolated with continuity of order 0, it is, howe-ver, difficult to directly use this model for optimizationpurposes. In [17] a simulation model based on the MOCand a continuously differentiable surface interpolation ofthe pump turbine characteristics [18] is applied for waterhammer studies. This modeling approach is also describedin [19]. A continuously differentiable interpolation of thepump turbine characteristics allows for the application ofnumerically efficient gradient based optimization methods.Therefore, a similar interpolation scheme is also providedin the present work, combining a state-of-the-art modelingapproach of the hydraulic system with a detailed model ofthe electric system, particularly modeling the energy los-ses of the electric subsystem in more detail compared toabove mentioned works.

Since PSPPs involve remarkable investments, suitablebuilding sites are limited and due to liberalized energymarkets, efficient operation is becoming increasingly im-portant. In particular, the additional degree of freedomgained from variable speed operation is utilized to incre-ase the efficiency of hydraulic machines. Reference [20]briefly describes the determination of optimal stationaryoperating points of a hydro power plant, where the hydrau-lic components and the generators are taken into account

in a special software tool. Stationary operating points ofmaximum efficiency are determined by an iterative pro-cess. Since the generators are modeled as active power-depending efficiencies, their operating constraints are notsystematically considered in the optimization process. In[4] control strategies for the optimized stationary opera-tion of variable speed pump turbines are suggested. Inturbine mode, the optimal set point of the turbine speedis determined for a given net head and a desired outputpower from a lookup table. Similarly, in pumping modean optimal guide vane opening is calculated for a givenrotational velocity and a given net head by a stationarylaw. In these cases, operating constraints are not explicitlytaken into account and in [4] optimality is solely based onthe pump turbine characteristics.

The mathematical model proposed in this paper is in-tended to serve as a basis for the analysis of the static anddynamic system behavior (e.g., water hammer studies),the design and test of control strategies and the determi-nation of optimal system operation. In this paper, the mo-del is used to calculate optimal stationary operating pointsof the PSPP for different modes of operation, which yieldminimal energy losses of the overall system while adheringto operating constraints.

The paper is organized as follows: Section 2 introducesthe considered variable speed PSPP and summarizes themathematical models of its components. In Section 3, thedynamic behavior of the PSPP is analyzed by simulationstudies. Section 4 describes the optimization problem foroptimal stationary operating points and presents its nume-rical results. Finally, Section 5 gives some conclusions.

2. Mathematical Model

The overall system, depicted in Fig. 1, consists of twoplant units A and B, which are coupled by a commonpipe system. Assuming constant grid voltage amplitudeand frequency, the coupling over the common electric gridsegment can be neglected. Further, it is assumed thatboth plant units have an identical configuration, which al-

2



lows to apply the same mathematical model to both plantunits. If necessary, the superscripts A and B are utili-zed to distinguish the different plant units. Subsequently,quantities in per unit representation are often used, whichwill be marked by (·).

2.1. Electric Subsystem

The equations of the electric system are given in aper unit dq-representation, normalized with respect tothe stator of the DFIM. The reference frame is synchro-nously rotating with the constant grid frequency ωgrid =2π50 rad/s, which is also used as the base frequency ωb inthe per unit representation, i.e. ωb = ωgrid, see Table 1.The respective 0-components are not taken into conside-ration, since they do not influence the dynamic behaviorof the system. The base values of power Sb and voltage ubfor the per unit representation of the electric subsystemare derived from the rated stator values of the DFIM, seeTable 1. The other base quantities are selected in accor-dance to [21], e.g., the base current and the base torqueread as

ib =2

3

Sb

uband Tb =

nppSb

ωb, (1)

see Table 1, with the number of pole pairs npp = 7.

Table 1: Base quantities.

Description Parameter Value Unit

apparent power Sb 182.50 MVA

peak value of phase voltage ub 15√

23

kV

phase current ib 9.93 kA

grid frequency ωb 2π50 rad/s

torque Tb 4.07 MNm

The electric grid is modeled as a balanced three-phasevoltage source of constant amplitude ugrid = 1 and, asalready mentioned before, a constant angular frequencyωgrid = 2π50 rad/s. The corresponding dq-components ofthe grid voltage are constant and coincide with the dq-components uSd1, uSq1 of the step-up transformer, see Fig.1. Using proper alignment between grid voltage and thereference frame, these voltages can be written as

uSd1 = ugrid = 1, uSq1 = 0. (2)

The electric system of each plant incorporates a DFIM (D),a step-up transformer (S), a converter transformer (C) anda voltage source converter (VSC), see Fig. 1. The trans-formers are necessary to adjust the corresponding voltagelevels. As discussed before, the DFIM can operate bothin generator mode (turbine mode) and motor mode (pum-ping mode). To allow for operation in both directions ofrotation, switching facilities are incorporated to reversethe phase order at the stator terminals, see, e.g., [22]. Thestart-up of the DFIM and the transition between pumpingand turbine mode require additional measures, both in the

electric and the hydraulic system, see, e.g., [23, 24]. Theseoperations are, however, not considered in this contribu-tion and thus not further discussed.

A magnetically linear model, which includes the influ-ence of iron losses, is used to describe the dynamic be-havior of the DFIM, see, e.g., [25, 26]. The differentialequations for the stator flux components ψD

d1 and ψDq1 read

as

1

ωb

dψDd1

dt= −RD

1 iDd1 + ωD

synψDq1 + uDd1 (3a)

1

ωb

dψDq1

dt= −RD

1 iDq1 − ωD

synψDd1 + uDq1, (3b)

with the stator resistance RD1 , the components iDd1 and iDq1

of the stator currents, and the components uDd1 and uDq1 of

the stator voltages. Moreover, ωDsyn is equal to the sign of

the rotation frequency ω of the rotor, i.e. ωDsyn = sign(ω).

The rotor fluxes ψDd2 and ψD

q2 are given in an equivalentform

1

ωb

dψDd2

dt= −RD

2 iDd2 +

(ωDsyn − ω

)ψDq2 + uDd2 (4a)

1

ωb

dψDq2

dt= −RD

2 iDq2 −

(ωDsyn − ω

)ψDd2 + uDq2. (4b)

Here, RD2 is the rotor resistance, iDd2 and iDq2 denote the

rotor current components, and uDd2 and uDq2 are the corre-sponding rotor voltage components. The dynamic modelof the DFIM is completed by the differential equations,see, e.g., [25, 26]

1

ωb

dψDdm

dt= −RD

i

(iDdm − iDd1 − iDd2

)+ ωD

synψDqm (5a)

1

ωb

dψDqm

dt= −RD

i

(iDqm − iDq1 − iDq2

)− ωD

synψDdm, (5b)

describing the magnetization flux components ψDdm and

ψDqm as functions of the magnetization current components

iDdm and iDqm and the stator and rotor current components.

Therein, the resistance RDi is used to represent the iron

losses of the DFIM. The fluxes of the DFIM are connectedwith the currents in the form

ψDd1 = ψD

dm + LD1 i

Dd1 ψD

q1 = ψDqm + LD

1 iDq1

ψDd2 = ψD

dm + LD2 i

Dd2 ψD

q2 = ψDqm + LD

2 iDq2

ψDdm = LD

miDdm ψD

qm = LDmi

Dqm,

(6)

with the leakage inductances LD1 and LD

2 of the stator androtor, respectively, and the magnetization inductance LD

m.Finally, the torque T of the DFIM reads as

T = LDm(iDd2i

Dqm − iDq2iDdm). (7)

The mathematical models of the converter transformer(superscript C) and the step-up transformer (superscript

3



S) are directly obtained from the model of the DFIM (3)-(6) by setting ω = 0, ωS

syn = ωCsyn = 1 and exchanging

superscripts. Different to the stator and rotor windings ofthe machine, the transformer windings are connected inthe vector group Yd5. This is incorporated into the modelby multiplying the voltages and currents of the secondaryside by appropriate constant rotation matrices, see, e.g.,[27].

The frequency converter is the essential component toallow for a variable speed operation of the pumped storageplant. There are a number of possible converter topologiesavailable in the medium voltage range and for the requiredpower ratings of approx. 20 MVA [28]. These converter to-pologies certainly have large differences in their setup andthe control strategies used to operate them. However, theobjective of the mathematical model derived in this pa-per is to describe the essential dynamic behavior of theoverall pumped storage plant. Thus, a rather generic ap-proach, which is capable of describing the average behaviorof many converter topologies, is used. It is an extensionof the so-called fundamental frequency or pseudo continu-ous converter model, see, e.g., [4, 29] and based on thefollowing assumptions:

• a back-to-back voltage source converter topology isused and

• the switching rate of the VSC is much faster than theremaining dynamics of the electric system.

The converter allows to directly assign the average voltagecomponents uCd2, uCq2 at the secondary side of the converter

transformer and uDd2, uDq2 at the rotor of the machine withincertain limits.

Simplifying the results in [30], the losses of the grid sideinverter (GSI) and the rotor side inverter (RSI) are ap-proximated by

PGSI = kGSI,0 + kGSI,1iC2 + kGSI,2

(iC2)2

(8a)

PRSI = kRSI,0 + kRSI,1iD2 + kRSI,2

(iD2)2

(8b)

with the current magnitudes

iC2 =

√(iCd2)2

+(iCq2)2

and iD2 =

√(iDd2)2

+(iDq2)2, (9)

and estimated parameters kGSI,j , kRSI,j , j = 0, 1, 2. The

power PC2 and PD

2 transferred by the converter to thetransformer and the DFIM, respectively, read as

PC2 = uCd2i

Cd2 + uCq2i

Cq2 and PD

2 = uDd2iDd2 + uDq2i

Dq2. (10)

Using the balance of energy, the dynamics of the DC-linkvoltage uDC can be written in the form

1

ωb

du2DC

dt= − 3

CDC

(PC2 + PGSI + PD

2 + PRSI

), (11)

with the effective DC-link capacitance CDC .

The mathematical model of the electric system is com-pleted by the electric interconnection of the transformatorsand the DFIM in form of the following constraints, see Fig.1:

1. The voltages at the common node are equal, i.e., interms of phase voltages of the three-phase system,

uSa2uSb2uSc2

=

uDa1uDb1uDc1

or

uSa2uSb2uSc2

=

uDa1uDc1uDb1

(12)

holds, depending on the switching state at the statorterminals of the DFIM. Neglecting the 0-components,this reads as

[uSd2uSq2

]=

[uCd1uCq1

]= K

[uDd1uDq1

](13)

in dq-representation, with K = diag[1, sign(ω)].

2. The sum of the currents at the common node is zero[iSd2iSq2

]+

[iCd1iCq1

]+ K

[iDd1iDq1

]=

[00

], (14)

which implies that only four of the six currents areindependent. In the following, iSd2 and iSq2 are chosenas the dependent currents.

The resulting mathematical model of the interconnectedelectric components can be represented in the form

1

ωb

dxe

dt= ge (xe, ω,ue,ugrid) (15)

with

xTe =

[iSd1, i

Sq1, i

Sdm, i

Sqm, i

Cd1, i

Cq1, i

Cd2, i

Cq2, (16a)

iCdm, iCqm, i

Dd1, i

Dq1, i

Dd2, i

Dq2, i

Ddm, i

Dqm, uDC

],

uTe =

[uCd2, u

Cq2, u

Dd2, u

Dq2

]. (16b)

The corresponding active and reactive power output ofthe pumped storage plant to the grid Pgrid, Qgrid are de-fined at the secondary side of the step-up transformer by1

Pgrid = −uDd1iDd1 − uDq1iDq1 − uCd1iCd1 − uCq1iCq1 (17a)

Qgrid =(uDd1i

Dq1 − uDq1iDd1

)sign(ω) + uCd1i

Cq1 − uCq1iCd1.

(17b)

2.2. Hydraulic Subsystem

The hydraulic system of the pumped storage powerplant consists of two reservoirs which are connected to twoFrancis turbines via a hydraulic pipe system, see Fig. 1.The hydraulic system is responsible for a major part of the

1In these equations, the voltages are functions of the system statexe, i.e. uDd1 = uDd1(xe), uDq1 = uDq1(xe), uCd1 = uCd1(xe) and uCq1 =

uCq1(xe).

4



dynamics and losses of the system. Thus, special emphasisis laid on an accurate modeling of the hydraulic compo-nents.

The central hydraulic component is the Francis turbine,which operates both in pumping and turbine mode. Sincethe transients of the pressures and volume flows inside theturbine are considerably faster than the dynamics of theremaining hydraulic system, it is possible to describe itsbehavior by a quasi-stationary characteristics, see, e.g.,[16, 31, 32]. It is common practice in literature to representthe turbine characteristics by using the specific speed N11,the specific volume flow q11 and the specific torque T11,which are related to the physical quantities in the form

N11 =NDr√Hn

, q11 =qPT

D2r

√Hn

, T11 =TPT

D3rHn

, (18)

see, e.g., [4, 33]. Here, N is the rotational speed, qPT isthe volume flow, TPT describes the torque and Dr is thereference diameter of the turbine. Further, the net headHn (also called dynamic head) is defined as

Hn =pUS − pDS

ρg+ ∆z +

v2US − v2DS

2g, (19)

with the fluid mass density ρ, the gravitational accelera-tion g = 9.81 m2/s, the upstream pressure pUS , the do-wnstream pressure pDS , the corresponding fluid veloci-ties vUS and vDS as well as the height difference ∆z =zUS − zDS between the inlet and the outlet of the turbine,see, e.g., [31]. Fig. 2 shows the measured characteris-tic maps of the Francis turbine as a function of the guidevane opening χ. Here, the quantities are normalized to theoperating point of best turbine efficiency, which is charac-terized by χr, N11,r, q11,r, T11,r.

The S-shape characteristics in the transition from tur-bine mode (N11, q11, T11 > 0) to the inverse pumping mode(N11 > 0, q11, T11 < 0) is problematic for the interpolationof the data points in a simulation model due to ambiguousoperating points for the same values of χ and N11. To ci-rcumvent this problem, the polar coordinates r and θ areintroduced in the form

r2 =

(N11

N11,r

)2

+

(q11q11,r

)2

=Hn,r

Hnf(N, qPT ), (20a)

θ = arctan

( q11q11,rN11

N11,r

)= arctan

( qPT

qPT,r

NNr

), (20b)

see, e.g., [16]. Here, the abbreviation f(N, qPT ) reads as

f(N, qPT ) =

(N

Nr

)2

+

(qPT

qPT,r

)2

, (21)

and the reference values qPT,r, Hn,r, TPT,r can be calcula-ted from the best efficiency point by (18) for a given valueof N = Nr. In this work, Nr is chosen2 as the synchronousrotational speed of the DFIM, see Table 2 for a definition

−2.0−1.5−1.0−0.5

0.00.51.01.5

S-shape

q 11/

q 11,

r

−1.5 −1 −0.5 0 0.5 1 1.5 2−4.0−3.0−2.0−1.0

0.01.02.0

N11/N11,r

T11/T

11,r

χ = 0.00 χ = 0.06χ = 0.17 χ = 0.33χ = 0.55 χ = 0.76χ = 0.95

Figure 2: Normalized characteristic maps of the Francis turbine,parametrized by the guide vane opening χ.

Table 2: Pump turbine parameters.

Parameter Value Unit

N11,r 81.37 rpm

q11,r 0.20 m3/s

T11,r 211.05 N m

χr 0.55

Nr 428.57 rpm

Hn,r 355.53 m

qPT,r 47.59 m3/s

TPT,r 3.44 MNm

Dr 3.58 m

∆z 11.50 m

of the pump turbine parameters. Using these new coordi-nates allows to calculate the transformed charactistic mapsin the form

WH =1

r2=

Hn

Hn,r

f(N, qPT )(22a)

WB = WHT11T11,r

=

TPT

TPT,r

f(N, qPT ), (22b)

see, e.g., [16, 31], which are shown in Fig. 3. While thistransformation eliminates the problem with ambiguousoperating points, it is no longer possible to represent thecase of a fully closed guide vane χ = 0 by the transformed

2Instead of choosing Nr, it would also be possible to choose Hn,r,e.g., as the nominal value of the net head.

5



characteristic maps. This case is, however, only of minorpractical interest and thus not considered in this paper. Asalready discussed before, an at least continuously differen-tiable approximation of the characteristic maps is requiredfor the simulation model and for the usage in gradient ba-sed optimization strategies. In this work, cubic B-Splinesurface interpolation based on surface skinning [34] wasapplied. The resulting interpolated characteristic mapsare given together with the measured data points in Fig.3.

−0.50.51.52.53.54.5

0.00.20.40.60.81.0

0

10

20

30

θχ

WH

χ = 0.06 χ = 0.17χ = 0.33 χ = 0.55χ = 0.76 χ = 0.95

−0.50.51.52.53.54.5

0.00.20.40.60.81.0

−202468

10

θχ

WB

Figure 3: Transformed pump turbine data points from Fig. 2 andinterpolated characteristic maps WH(χ, θ) and WB(χ, θ).

Using (22), the net head Hn and the pump turbine tor-que TPT

Hn(χ,N, qPT ) = Hn,rWH(χ, θ(N, qPT ))f(N, qPT ),(23a)

TPT (χ,N, qPT ) = TPT,rWB(χ, θ(N, qPT ))f(N, qPT )(23b)

are defined for given volume flow qPT , rotary speed N andguide vane position χ of the pump turbine.

Note that in [18] the specific quantities N11, q11 andT11 are interpolated on B-Spline surfaces, which allows torepresent fully closed guide vanes. In some circumstances,however, it might be considered a drawback, that insteadof θ a more general parametrization variable has to beapplied, which doesn’t have an immediate relationship toany physical variable of the system.

The pump turbines of the power plant are connected tothe upper and lower reservoir by an arrangement of pipesegments, cf. Fig. 1. Assuming a one-dimensional axialflow, a single pipe segment is modeled by the PDEs for the

pressure p and the fluid velocity v = qA in the form, see,

e.g., [31, 35, 36, 37]

1

ρa2

(∂p

∂t+ v

∂p

∂x

)+

1

A

∂A

∂xv +

∂v

∂x= 0, (24a)

1

ρ

∂p

∂x+

(∂v

∂t+ v

∂v

∂x

)+λv |v|2D

+ g sin(α) = 0, (24b)

where a is the wave speed, A(x) is the cross sectional area,D(x) the corresponding diameter, α the inclination of thepipe segment and λ the Darcy-Weissbach friction parame-ter. Table 3 summarizes the parameters of the hydraulicpipe system of the power plant. It is assumed that the dia-meter D(x) of a pipe element varies linearly with the posi-tion x along the pipe, i.e. D(x) = D(0)+(D(l)−D(0))x/l,with the diameters D(0) and D(l) corresponding to theareas A(0) and A(l) at the beginning and the end of thepipe segment, respectively.

Table 3: Parameters of the pipe system according to Fig. 1.

ili

lmax

Ai(0)Amax

Ai(l)Amax

αi ai λi × 103

1 2.0× 10−1 1.00 1.00 −3.7◦ 1250 m/s 5.3

2 6.2× 10−3 1.00 0.85 −3.7◦ 1250 m/s 5.3

3 3.8× 10−1 0.85 0.85 −90◦ 1250 m/s 5.3

4 5.9× 10−3 0.85 0.43 0◦ 1200 m/s 5.3

5 9.7× 10−2 0.43 0.43 0◦ 1200 m/s 5.3

6 3.0× 10−3 0.21 0.23 0◦ 1200 m/s 5.8

7 2.7× 10−2 0.23 0.23 0◦ 1200 m/s 5.8

8 4.9× 10−2 0.32 0.32 6.1◦ 1250 m/s 5.8

9 3.1× 10−3 0.32 0.21 6.1◦ 1250 m/s 5.8

10 4.0× 10−2 0.43 0.43 6.1◦ 1250 m s 5.3

11 6.2× 10−3 0.43 1.00 6.1◦ 1250 m/s 5.3

12 1.0 1.00 1.00 5.6◦ 1250 m/s 5.3

The interconnection of the pipe segments and the con-nection with the pump turbine or the reservoirs is repre-sented by suitable boundary conditions. Assuming thatthe filling levels of the reservoirs are approximately con-stant during the considered short time scales of the dyna-mics simulations, the upper and the lower reservoir intro-duce the following boundary conditions

p1(0) = pUR and p12(l12) = pLR, (25)

with the pressures pUR and pLR at the inlets of the upperand lower reservoir.

The lossless connection of two pipe segments i and i +1 with differing parameters is modeled by the boundaryconditions

pi(li, t) = pi+1(0, t), (26a)

qi(li, t) = qi+1(0, t), (26b)

i = 1, . . . , 11. The boundary conditions for a losslessjunction of the pipe segment i into the two pipe segments

6



j and k are given by

pi(li, t) = pj(0, t) = pk(0, t), (27a)

qi(li, t) = qj(0, t) + qk(0, t), (27b)

and the boundary conditions for the junction of pipe seg-ments j and k to pipe segment i read as

pi(0, t) = pj(lj , t) = pk(lk, t), (28a)

qi(0, t) = qj(lj , t) + qk(lk, t). (28b)

Finally, the connection of a pump turbine to the up-stream pipe segment i and the downstream pipe segmentj is described by the boundary conditions

pi(li, t) = pj(0, t) + ∆pPT (29a)

qi(li, t) = qj(0, t) = qPT , (29b)

where the pressure drop ∆pPT = pUS − pDS is defined asa function of the volume flow qPT , the guide vane positionχ and the rotary speed N of the pump turbine by

∆pPT = ρg

(Hn −∆z − q2PT

1(Ai(li))2

− 1(Aj(0))2

2g

), (30)

cf. (19). Here, the net head Hn = Hn(χ,N, qPT ) is definedby (23a).

In conclusion, the overall hydraulic part of the powerplant is represented by a set of coupled PDEs, which needsto be numerically solved for a dynamic simulation of thepower plant. For this task, it is useful to apply a numberof simplifications to (24): (i) In the present application,the fluid velocity is much smaller than the wave speed, i.e.v � a, which allows to neglect the convective terms v ∂p

∂x

and v ∂v∂x in (24), see, e.g., [37]. (ii) It is further feasible

to assume a constant mass density ρ = 1000 kg/m3 and aconstant wave speed3 a. This yields a simplified PDE ofthe form

1

ρa2∂p

∂t+

1

A

∂A

∂xv +

∂v

∂x= 0, (31a)

1

ρ

∂p

∂x+∂v

∂t+λv |v|2D

+ g sin(α) = 0, (31b)

which can be numerically solved by applying the well-established MOC, see, e.g., [31]. To do so, the simplifiedPDE (31) is transformed to

dp

dt± ρadv

dt+ρa2

A

∂A

∂xv ± ρa

(λv |v|2D

+ g sin (α)

)= 0,

(32)

which is defined on the characteristics

dx

dt= ±a. (33)

To solve this set of differential equations, a discretizationin time and space is used in the subsequent simulations,see, e.g., [31].

3Note that the wave speed summarizes the effects of the fluidcompressibility and the pipe wall elasticity.

2.3. Drive Train

The drive train couples the turbine with the generatorand thus the hydraulic with the electric subsystem. Usingthe base quantities in Table 1, the balance of momentumfor the drive train in per unit representation is given by

2Hdω

dt=T + TPT − dc sign(ω)− dqω |ω| , (34)

with the inertia constant H, see, e.g., [21], comprising theinertia of the pump turbine and the rotor of the DFIM.The torque T of the DFIM and the torque TPT are given by(7) and the per unit representation of (23b), respectively.The mechanical losses of the drive train are split into aCoulomb part dc sign(ω) and a quadratic part due to ven-tilation losses dqω |ω|. Table 4 summarizes the essentialparameters of the mechanical system.

Table 4: Parameters of the mechanical system.


H 3.86 s

dc 4.12× 10−3

dq 6.13× 10−3

3. Transient Simulation

As it is discussed in the introduction, the mathematicalmodel developed in the previous section serves two pur-poses: (i) First, it is capable of accurately simulating thetransient system behavior. Thus, it is suitable for an ana-lysis of the dynamic system behavior and for testing diffe-rent control and operating strategies in simulation studies.(ii) Second, the mathematical model also serves as a ba-sis for reduced models, which are used for the design ofcontrol and estimation strategies. In this context, optimaloperating and control strategies, which allow for energy-optimal (dynamic) operation of the power plant, are ofparticular interest.

In the next section, a simplified model is derived andoptimal stationary operating points are calculated. In thissection, some main features of the pumped storage po-wer plant are discussed using transient simulations of themathematical model. For this purpose, the model is im-plemented in Matlab/Simulink. The MOC is appliedfor the numerical solution of the PDEs of the pipe system,using a total of 594 spatial discretization points.

At first, it is assumed that both plant units A and B areoperated identically. In the following simulation results,values of volume flow and pressure are depicted in per unitrepresentation. Therefore, the base volume flow qb = qPT,r

and the base pressure pb = Sb/qb are defined, see Table 5.Fig. 4 depicts simulation results of the open-loop beha-

vior of the power plant for filtered (first order delay with100 ms rise time) step-like changes of the inputs uT

e =

7



Table 5: Base quantities of the hydraulic subsystem.


qb 47.59 m3/s

pb 38.35 bar

[uCd2, uCq2, u

Dd2, u

Dq2] of the electric system and χ of the hy-

draulic system. The input values in Fig. 4(a) correspond tostationary operating points with PA

grid = PBgrid = 160 MW

and PAgrid = PB

grid = 55 MW active power, respectively,using a power factor of cos(ϕ) = 0.9 to calculate the cor-responding reactive power QA

grid = QBgrid.

−0.25

0.00

0.25

0.50

0.75

1.00

Inpu

ts

(a)

χ uDd2 uD

q2 uCd2 uC

q2

0.0

0.5

1.0

1.5

2.0

2.5

Mec

hani

calv

aria

bles

(b)

pUS pDS qPT w

−50

0

50

100

150

200

Pow

ers

inM

W

(c)

Pgrid Qgrid

0 10 20 30 40 500.75

1.00

1.25

1.50

Time in seconds

uD 1

(d)

Figure 4: Fast change of operating points from PAgrid = PB

grid =

160 MW to PAgrid = PB

grid = 55 MW: (a) System inputs, (b) mecha-

nical system variables, (c) active and reactive power and (d) magni-tude of the stator voltage uD1 .

Fig. 4(b) shows the upstream pressure pUS = p7(l7), thedownstream pressure pDS = p8(0), the volume flow qPT

and the rotational speed ω of the pump turbine. The rapid

reduction of the guide vane opening causes a fast decreaseof the volume flow qPT . Furthermore, large oscillationsare induced in the upstream and downstream pressures,which propagate through the pipe system as pressure wa-ves. The peak value of the upstream pressure exceeds thestationary pressure by almost 35%, which poses a possi-ble threat to the pipe’s mechanical integrity. A similarproblem occurs for the downstream pressure which almostreaches zero. In this case, cavitation can occur, which alsoresults in large stress of the pipe system and the turbine.These simulation results show that water hammer effectsare immanent hazards to pumped storage power plants iffast changes between operating points occur.

In Fig. 4(c) the active and reactive output power of oneplant unit are shown. The pronounced oscillating behaviorafter changing the operating point indicates a significantlyfaster dynamics of the electric system in comparison to thehydraulic and mechanical system. This fact can be advan-tageously used for the design of a control strategy, where,e.g., a cascaded control of the electric and hydraulic sy-stem is employed. Note that the oscillations in the electricsystem can be easily suppressed by a suitable control ofthe inverter of the DFIM. Finally, the magnitude of thestator voltage

uD1 =

√(uDd1)2

+(uDq1)2

(35)

is depicted in Fig. 4(d), which also shows large variationsduring the fast transition. Fig. 5 illustrates this transitionin the pump turbine characteristic map.

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.5

1

1.5

χ = 0.06

χ = 0.17χ = 0.33

χ = 0.55χ = 0.76χ = 0.95

N11/N11,r

q 11/

q 11,

r

TrajectoryStartEnd

Figure 5: Fast change of operating points from PAgrid = PB

grid =

160 MW to PAgrid = PB

grid = 55 MW: Trajectory in the pump turbine

characteristics.

To point out the strong dynamic coupling between theplant units, the simulation is repeated applying the step-like changes in Fig. 4(a) only to unit A, while keepingthe inputs of unit B constant. Fig. 6 reveals that thepressure waves initiated at pump turbine A travel along topump turbine B, driving it away from its initial stationaryoperating point. At the end of the simulation horizon, thesystem is in a condition of slowly decaying oscillations.Apparently, in the case of independent operation, the plantunits can cause remarkable mutual disturbances, whichhave to be thoroughly treated by a control strategy.

Large oscillations in the pressures and the active and re-active power are inadmissible in practical application and

8



0.0

0.5

1.0

1.5

2.0

2.5

p US

and

q PT

(a)

pAUS pB

US qAPT qB

PT

0 10 20 30 40 500.0

0.5

1.0

1.5

Time in seconds

p DS

andω

(b)

pADS pB

DS ωA ωB

Figure 6: Fast change of operating points for independent operatingplant units: (a) Upstream pump turbine pressure pUS and pumpturbine volume flow qPT of units A and B, (b) Downstream pumpturbine pressure pDS and angular velocity ω of units A and B.

are prevented by limiting the rate of change of, e.g., theguide vane position χ in existing pumped storage powerplants. This, on the other hand, also limits the rate ofchange of the output power, which is a major drawbackif the power plant should be used for the stabilization offast fluctuations in the grid. An accurate model which cor-rectly predicts the water hammer phenomena in the pipesystem, as the one presented in this paper, is thus indis-pensable for the design and test of control strategies whichallow for a fast reaction to fluctuations in the distributionnetwork.

4. Optimal Stationary Operating Points

The presented dynamical model can be used as the basisfor dynamic simulation studies, the design of control andobserver strategies and the calculation of optimal statio-nary and dynamic operating strategies. In this section, op-timal stationary operating points are calculated for givenvalues of the active and reactive power, Pgrid and Qgrid,respectively, based on a minimization of the overall sy-stem losses. To do so, a stationary solution of the pipe sy-stem model is determined in the subsequent section. Thisstationary pipe model in combination with the stationaryequations of the electric system serves as the basis for theformulation of the optimization problem to minimize thesystem losses.

4.1. Stationary Solution of the Pipe System

If the time derivative ∂p/∂t in (31a) is set to zero, it isimmediately clear that the volume flow q(x) is independentof x, i.e. q(x) = q = const.. Using this result and ∂q/∂t =

0 in (31b), and integration over the pipe length l, givesthe pressure drop p(l)− p(0) over this pipe segment in theform

p(l) = p(0)− q(l)2ρ1

(A(l))2− 1

(A(0))2

2

− q(l) |q(l)| ρλ2

∫ l

0

1

D(x) (A(x))2 dx− ρgl sin(α).

(36)

Application of this stationary solution to the equations ofthe interconnected pipe system gives, after some calcula-tions, the stationary solution of the overall pipe system inthe form

0 = ρg(Hg −Hn(χA, NA, qAPT )) +(qAPT

)2c0

− (qAPT + qBPT )∣∣qAPT + qBPT

∣∣ c1 − qAPT

∣∣qAPT

∣∣ c2(37a)

0 = ρg(Hg −Hn(χB , NB , qBPT )) +(qBPT

)2c0

− (qAPT + qBPT )∣∣qAPT + qBPT

∣∣ c1 − qBPT

∣∣qBPT

∣∣ c2,(37b)

where A and B refer to the two pump turbines of thepumped storage power plant. The overall gross head Hg

is given by

Hg =pUR − pLR

ρg+ ∆z −

12∑

i=1

li sin(αi)

and the coefficients c0, . . . , c2 in (37) are defined as

c0 =ρ

2

((1

AUS

)2

−(

1

ADS

)2)

c1 =ρ

2

∑

i∈Icλi

∫ li

0

1

Di(x) (Ai(x))2 dx

c2 =ρ

2

∑

i∈Ibλi

∫ li

0

1

Di(x) (Ai(x))2 dx.

Therein, the set Ic refers to the pipes which are commonto both pump turbine units, Ic = {1, . . . , 5, 10, . . . , 12}and Ib summarizes the branched pipe segments, i.e., Ib ={6, . . . , 9}.

The hydraulic input power for the turbine operation isdefined as

Phyd,in = ρg(qAPT + qBPT

)Hg +

((qAPT

)3+(qBPT

)3)c0.

(39)Utilizing the stationary solution (37), the balance of hy-draulic power can be formulated as Phyd,in = PA

in + PBin +

Pdiss, with the hydraulic input power PAin and PB

in of pumpturbine A and B, respectively, and the dissipated power

9



Pdiss due to the friction in the pipe system.

PAin = ρgqAPTHn

(χA, NA, qAPT

)(40a)

PBin = ρgqBPTHn

(χB , NB , qBPT

)(40b)

Pdiss = (qAPT + qBPT )2|qAPT + qBPT |c1 (40c)

+((qAPT

)2 ∣∣qAPT

∣∣+(qBPT

)2 ∣∣qBPT

∣∣)c2

4.2. Constrained Nonlinear Optimization Problem

In this section, optimal stationary operating points forthe pumped storage power plant comprising two pump tur-bines, which share a common pipe system, are calculated.The main goal is to minimize the losses Pl of the over-all hydraulic and electric system. The stationary systemlosses consist of

• pipe friction losses,

• hydraulic losses in the pump turbines,

• mechanical friction and ventilation losses and

• electric losses including losses of the VSCs as well ascopper and iron losses of the converter transformersand the DFIMs.

A per unit representation of the complete system is appliedfor the numerical treatment of the optimization problem,using the respective base quantities, see Tables 1 and 5.

Instead of summarizing all the individual losses, theoverall losses can be formulated as the difference of thehydraulic input power Phyd,in and the power supplied to

the distribution network PAgrid + PB

grid, i.e.

Pl(x,u) = Phyd,in − PAgrid − PB

grid. (41)

Therein, the input and output power are (nonlinear)functions of the system state x and system input u

xT =[qAPT ωA

(xAe

)TqBPT ωB

(xBe

)T] (42a)

uT =[χA

(uAe

)TχB

(uBe

)T] . (42b)

In addition to minimizing the overall system losses, thestationary operating points should be calculated such thatthe active and reactive power P j

grid and Qjgrid, j ∈ {A,B},

supplied to the grid by the two DFIMs are equal to thedesired values P j∗

grid and Qj∗grid. This is considered in the

optimization task by the equality constraints

gj1(x,u) = P jgrid − P

j∗grid = 0 (43a)

gj2(x,u) = Qjgrid − Q

j∗grid = 0, (43b)

j ∈ {A,B}. Furthermore, in order to reduce the lossesof the grid side inverter, it is common practice to set thereactive power QCj

1 at the converter transformers to zero,i.e.

gj3(x,u) = uCjq1 i

Cjd1 − u

Cjd1 i

Cjq1 = 0, j ∈ {A,B}. (43c)

Finally, a stationary operating point is obtained by (37)and by setting the right hand sides of (15) and (34) tozero.

For a stationary operating point to be feasible, the con-straints of the real system have to be met. The guide vaneposition χj is limited by χmin ≤ χj ≤ χmax, which can beequivalently formulated by

hj1(x,u) = χj − χmax ≤ 0 (44a)

hj2(x,u) = −χj + χmin ≤ 0, (44b)

for j ∈ {A,B}. The stator and rotor current amplitudesof the DFIM are limited due to thermal constraints in theform

hj3(x,u) =(iDjd1

)2+(iDjq1

)2−(iDj1,max

)2≤ 0 (44c)

hj4(x,u) =(iDjd2

)2+(iDjq2

)2−(iDj2,max

)2≤ 0 (44d)

and the rotor voltage amplitude is basically limited by theDC voltage of the VSC

hj5(x,u) =(uDjd2

)2+(uDjq2

)2−(uDj2,max

)2≤ 0, (44e)

j ∈ {A,B}. Finally, the rotor active power is limited dueto the thermal constraints of the VSC by

hj6(x,u) = uDjd2 i

Djd2 + uDj

q2 iDjq2 − PD

2,max ≤ 0 (44f)

hj7(x,u) = −uDjd2 i

Djd2 − u

Djq2 i

Djq2 − PD

2,max ≤ 0. (44g)

The respective limits are summarized in Table 6.

Table 6: Limits for the inequality constraints (44).

Parameter Value

χmin 0.06

χmax 0.95

PD2,max 0.082

iD1,max 1.000

iD2,max 1.346

uD2,max 0.121

In order to avoid results outside of valid area of theinterpolated pump turbine characteristic maps, cf. Fig. 3,the auxiliary inequality constraints

hj8(x,u) = P jout − P j

in ≤ 0, (45)

j ∈ {A,B}, are introduced. Therein P jout = ωj T j

PT and

P jin represent the per unit mechanical output power and

the per unit hydraulic input power, cf. (40a) and (40b),of the respective pump turbines.

Putting together the equality and inequality constraints,the task of calculating optimal stationary operating points

10



is formulated as the constrained nonlinear optimizationproblem

minu

Pl(x,u)

s.t. g(x,u) = 0

h(x,u) ≤ 0.

(46)

This optimization problem is numerically solved by a gra-dient based method using the fmincon command of Mat-lab. The utilization of a B-Spline interpolation of thepump turbine characteristics of Fig. 3 allows to provideanalytical expressions for all required gradients, which sig-nificantly improves the accuracy and speed of the numeri-cal solution.

In the following, optimal stationary operating points arepresented for four scenarios: (i) In the first scenario, bothplant units are operated identically, i.e. with the sameinputs, states and output power, which is a possible ope-rating mode when the two plant units are owned by onecompany. (ii) For small values of the desired grid power,the results of (i) are compared with the optimal operationusing only a single plant unit, while the other plant unitis switched off. (iii) Typically, both plant units distributetheir power to the same transmission line. Thus, the thirdoperating scenario examines if there are any benefits of acoordinated operation of the two plant units. (iv) Finally,the advantages of a variable speed system in view of theoverall efficiency of the plant and the operating range isexamined in the fourth scenario.

It has to be noted that the optimal operating pointsare strongly influenced by the interpolated characteristicmaps WH(χ, θ) and WB(χ, θ), cf. Fig. 3. Therefore, thereliability of the optimization results strongly depends onthe accuracy of the pump turbine characteristics.

4.3. Identical Operation of Plant Units

In this first scenario, it is assumed that both plant unitsare operated identically in turbine mode, distributing thesame active power PA

grid = PBgrid to the grid. A series of 200

optimal stationary operating points is calculated for threedifferent gross heads Hg = 275 m, Hg = 335 m and Hg =395 m. Here, the first value corresponds to the minimumgross head for almost empty upper reservoir and almostcompletely filled lower reservoir, while the last value isgiven for an almost completely filled upper reservoir andalmost empty lower reservoir. The range of the desiredactive power P ∗grid delivered to the distribution network isthen defined in the form

P ∗grid = 100 MW, . . . , 250 MW for Hg = 275 m

P ∗grid = 100 MW, . . . , 340 MW for Hg = 335 m

P ∗grid = 100 MW, . . . , 350 MW for Hg = 395 m,

with P ∗grid = PA∗grid + PB∗

grid. The desired reactive power

Q∗grid = QA∗grid + QB∗

grid > 0 is set according to the power

0.0

0.2

0.4

0.6

0.8

1.0

χ

(a)

Hg = 275 mHg = 335 mHg = 395 m

010203040506070

q PT

inm

3 /s

(b)

100 150 200 250 300 3500.85

0.90

0.95

1.00

1.05

1.10

Pgrid in MW

ω

(c)

Figure 7: Optimal stationary operating points for identical operationof the plant units: (a) guide vane opening χ, (b) volume flow of aplant unit qPT and (c) angular velocity ω.

factor cos(ϕ) = 0.9, which is the equivalent choice as hasbeen used in the dynamic simulations in Section 3.

The resulting values of the guide vane position χ, thevolume flow qPT and angular velocity ω are depicted inFig. 7. Fig. 7(b) reveals that smaller gross heads Hg im-ply higher volume flows in order to produce the same hy-draulic input power, cf. (39). This in turn leads to largerguide vane openings, as depicted in Fig. 7(a). Fig. 7(c)shows that the angular velocity of the plant units is gene-rally decreased for lower desired grid power. The minimumvelocity is, however, limited by the constraints of the elec-tric system described in (44c)-(44g), in particular by themaximum rotor voltage which can be supplied by the VSI,see also Fig. 9(b).

The overall system efficiency for the turbine mode

η =PAgrid + PB

grid

Phyd,in

(48)

is depicted in Fig. 8(a), showing differing points of maxi-mum efficiency for the three values of the gross head Hg.For all cases, however, a high overall system efficiency isobtained in a rather wide operating range of the desiredgrid power. To a significant degree, this can be attributedto the variable speed operation of the plants, which, inthe case of Hg = 335 m, will be shown in Section 4.5. The

11



100 150 200 250 300 3500.75

0.80

0.85

0.90

0.95

Pgrid in MW

η

(a)

Hg = 275 mHg = 335 mHg = 395 m

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.5

1

1.5

χ = 0.06χ = 0.17

χ = 0.33

χ = 0.55χ = 0.76χ = 0.95

N11/N11,r

q 11/

q 11,

r

(b)Hg = 275 mHg = 335 mHg = 395 m

Figure 8: Optimal stationary operating points for identical operationof the plant units: (a) overall efficiency η and (b) operating pointsin the pump turbine characteristics.

−0.20.00.20.40.60.81.0

volta

gein

puts

(a)

uDd2 uD

q2

uCd2 uC

q2

100 150 200 250 300 350−1.0

−0.5

0.0

0.5

1.0

Pgrid in MW

norm

aliz

edva

riab

les

(b)

iD1 /i

D1,max iD

2 /iD2,max

uD2 /u

D2,max PD

2 /PD2,max

Figure 9: Optimal stationary operating points for identical operationof the plant units and Hg = 335 m: (a) electric system inputs and(b) constrained electric variables.

operating points are represented in the characteristic mapof the Francis turbine in Fig. 8(b), where again the ope-rating area with constant angular velocity can be clearlyidentified.

Fig. 9(a) displays the electric system inputs, i.e. the ro-tor voltages uDd2, uDq2 and the grid side converter voltages

uCd2, uCq2, necessary to obtain the desired grid power for thecase Hg = 335 m. In Fig. 9(b), the corresponding constrai-

100 150 200 250 300 3500

10

20

30

40

50

60

Pgrid in MW

pow

erin

MW

overall pump turbinemechanical electricpipe friction

Figure 10: Optimal stationary operating points for identical ope-ration of the plant units and Hg = 335 m: analysis of the systemlosses.

ned variables of the electric system are shown. As alreadymentioned before, the magnitude uD2 of the rotor voltagesticks to its upper bound for Pgrid < 206 MW resulting inthe limitation of the angular velocity in Fig. 7(c).

Finally, Fig. 10 shows the contribution of the differentparts of the plant to the overall system losses. Obviously,the majority of the system losses results from the lossesin the pump turbine. On the other hand, with an over-all amount of approximately 10 MW, consideration of theremaining losses (mechanical, electric and pipe friction los-ses) is certainly important and relevant for the calculationof optimal operating points.

4.4. Independent Operation and Single Plant Unit

In the last subsection, it was assumed that both plantunits operate identically. The purpose of this subsectionis to analyze if an improvement of the overall system ef-ficiency can be achieved by an independent operation ofthe two plant units, which means that the assumptionPA∗grid = PB∗

grid is dropped in the calculation of optimal ope-rating points. Moreover, it is examined in which cases itis more efficient to operate only a single plant unit andto switch off the second plant unit. The subsequent op-timization results are calculated for Hg = 335 m, and theentire possible pumping and turbine mode of operation isconsidered. The corresponding maximum and minimumvalues of the grid power P ∗grid are limited by the systemconstraints. The desired reactive power Q∗grid > 0 is againset according to the power factor cos(ϕ) = 0.9.

Fig. 11(d) reveals that there is some room of impro-vements in terms of overall efficiency in the case of inde-pendent operation compared to the identical operation ofthe plant units, in particular for the pumping mode. Theseimprovements of efficiency primarily result from reducti-ons in pump turbine losses caused by asymmetric opera-ting points, cf. Fig. 11(a)-(c). A larger share of the totalpower is transmitted over a pump turbine unit that is ope-rated at higher efficiency. Furthermore, Fig. 11(d) showsthat it is far more efficient to use a single plant unit forsmall values of the grid power Pgrid, up to cases of almost

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0.0

0.2

0.4

0.6

0.8

1.0

χ

(a)

010203040506070

|q PT|in

m3 /

s

(b)

0.85

0.90

0.95

1.00

1.05

1.10

|ω|

(c)

indep. operation: unit Aindep. operation: unit B

−300 −200 −100 0 100 200 3000.70

0.75

0.80

0.85

0.90

0.95

Pgrid in MW

η

(d)

ident. op.indep. op.single unit

Figure 11: Optimal stationary operating points for independent ope-ration of the plant units and single plant unit: (a) guide vane openingχ, (b) volume flow of a single plant unit qPT , (c) angular velocity ωand (d) system efficiency η.

the maximum possible grid power of a single plant unit.This results from the fact that the dominating part of theoverall system losses are the pump turbine losses, suchthat it makes sense to choose operating points of the plantwhere the efficiency of the pump turbine is best. Giventhe results of Fig. 11(d) this is the case for large values ofthe guide vane opening.

4.5. Variable and Fixed Angular Speed Operation

In this final simulation scenario, the benefits of a va-riable angular speed operation compared to an operationwith fixed angular speed with respect to the overall systemefficiency is examined. For this scenario, again an identicaloperation of both plant units is considered.

Fig. 12(a) shows that the overall system efficiency canbe significantly increased by the variation of the angular

−300 −200 −100 0 100 200 3000.75

0.80

0.85

0.90

0.95

Pgrid in MW

η

(a)

variablefixed

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.5

1

1.5

χ = 0.06χ = 0.17

χ = 0.33

χ = 0.55χ = 0.76χ = 0.95

N11/N11,r

q 11/

q 11,

r

(b)

−1.4 −1.2 −1 −0.8 −0.6−1.5

−1

−0.5

0

0.5

χ = 0.06

χ = 0.17

χ = 0.33

χ = 0.55

χ = 0.76

χ = 0.95

N11/N11,r

q 11/

q 11,

r(c)

Figure 12: Optimal stationary operating points for variable and fixedangular speed operation: (a) overall system efficiency η, (b) and (c)operating points in the characteristic map of the pump turbine.

speed of the pump turbine, in particular in operating ran-ges with a power below the rated power of the plant. Mo-reover, the overall feasible operating range of the pumpturbine is increased by variable speed operation, whichis most visible in the pumping mode of the power plant.Fig. 12(b) and 12(c) show the operating points in the cha-racteristic map of the pump turbine.

Finally, Fig. 13 shows the values of the guide vane posi-tion, the volume flow of a single plant unit and the angularspeed corresponding to the optimal operating points. Itcan be deduced from these plots that in the pump modemore water can be pumped from lower reservoir to the up-per by the variable speed plant compared to a fixed speedplant, while using the same electric input power. Analo-gously, less water is necessary to generate the same powerusing a variable speed plant compared to a fixed speedplant. These results clearly show the advantages of usinga variable speed pumped storage plant system.

13



0.0

0.2

0.4

0.6

0.8

1.0

χ

(a)

variablefixed

0102030405060

|q PT|in

m3 /

s

(b)

−300 −200 −100 0 100 200 3000.85

0.90

0.95

1.00

1.05

1.10

Pgrid in MW

|ω|

(c)

Figure 13: Optimal stationary operating points for variable and fixedangular speed operation: (a) guide vane opening χ, (b) volume flowof single plant unit qPT and (c) angular speed ω.

5. Conclusions

In this contribution, a detailed mathematical model ofa variable speed pumped storage power plant is presented.The behavior of the electric and mechanical key compo-nents are considered to obtain a comprehensive descriptionof the overall dynamic system behavior. In particular, theinfinite-dimensional nature of the pipe system is properlyhandled by the application of the Method of Characteris-tics, accurately reproducing important dynamical effectslike wave propagation due to water hammer. This mathe-matical model can therefore serve as a basis for dynamicalsimulation studies and the development and test of (opti-mal) control strategies.

The proposed model is also utilized to calculate optimalstationary operating points of minimum energy losses. Themost important energy losses including pump turbine los-ses, pipe friction, mechanical friction as well as several elec-tric losses are systematically included in the model. Thenonlinear optimization problem also takes into account theconstraints of the system states. Optimization results fordifferent operating scenarios are presented, which clearlyshow the advantages of a variable speed plant comparedto a fixed speed plant. Moreover, some important conclu-sions on the optimal operation of variable speed pumped

storage power plants with double fed induction machinesare drawn from these optimization results.

Acknowledgements

This research was funded by the Austrian Research Pro-motion Agency (FFG) under grant agreement numbers833955 and 849267. The authors are grateful to the in-dustrial research partner Andritz Hydro GmbH.

References

[1] J. I. Perez-Dıaz, M. Chazarra, J. Garcıa-Gonzalez, G. Cavazzini,A. Stoppato, Trends and challenges in the operation of pumped-storage hydropower plants, Renewable and Sustainable EnergyReviews 44 (2015) 767–784.

[2] A. Hodder, Double-fed asynchronous motor-generator equippedwith a 3-level VSI cascade, Ph.D. thesis, EPFL, Lausanne, Swit-zerland (2004).

[3] Y. Pannatier, Optimisation des strategies de reglage d’une in-stallation de pompage-turbinage a vitesse variable, Ph.D. thesis,EPFL, Lausanne, Switzerland (2010).

[4] Y. Pannatier, B. Kawkabani, C. Nicolet, J. J. Simond,A. Schwery, P. Allenbach, Investigation of Control Strategiesfor Variable-Speed Pump-Turbine Units by Using a SimplifiedModel of the Converters, IEEE Transactions on Industrial Elec-tronics 57 (9) (2010) 3039–3049.

[5] E. Schmidt, J. Ertl, A. Preiss, R. Zensch, R. Schurhuber,J. Hell, Studies about the Low Voltage Ride Through Ca-pabilities of Variable-Speedpeed Motor-Generators of PumpedStorage Hydro Power Plants, in: 21st Australasian UniversitiesPower Engineering Conference (AUPEC), 2011, pp. 1–6.

[6] J. A. Suul, K. Uhlen, T. Undeland, et al., Variable speed pum-ped storage hydropower for integration of wind energy in iso-lated grids: case description and control strategies, in: NordicWorkshop on Power and Industrial Electronics (NORPIE), Es-poo, Finland, 2008.

[7] P. Steimer, O. Senturk, S. Aubert, S. Linder, Converter-FedSynchronous Machine for Pumped Hydro Storage Plants, in:Energy Conversion Congress and Exposition (ECCE), IEEE,Pittsburgh, PA, USA, 2014, pp. 4561–4567.

[8] H. Schlunegger, A. Thoni, 100 MW full-size converter in theGrimsel 2 pumped-storage plant, in: Hydro 2013, Innsbruck,Austria, 2013.

[9] A. Bocquel, J. Janning, Analysis of a 300 MW Variable SpeedDrive for Pump-Storage Plant Applications, in: European Con-ference on Power Electronics and Applications, Dresden, Ger-many, 2005, pp. 1–10.

[10] J. Lung, Y. Lu, W. Hung, W. Kao, Modeling and DynamicSimulations of Doubly Fed Adjustable-Speed Pumped StorageUnits, IEEE Transactions on Energy Conversion 22 (2) (2007)250–258.

[11] N. Sivakumar, D. Das, N. Padhy, Variable speed operation ofreversible pump-turbines at Kadamparai pumped storage plant- A case study, Energy Conversion and Management 78 (2014)96 – 104.

[12] J. Liang, R. G. Harley, Pumped Storage Hydro-Plant Models forSystem Transient and Long-Term Dynamic Studies, in: IEEEPES General Meeting, 2010, pp. 1–8.

[13] J. I. Sarasua, J. I. Perez-Dıaz, J. R. Wilhelmi, J. A. Sanchez-Fernandez, Dynamic response and governor tuning of a longpenstock pumped-storage hydropower plant equipped with apump-turbine and a doubly fed induction generator, EnergyConversion and Management 106 (2015) 151–164.

[14] H. M. Gao, C. Wang, A Detailed Pumped Storage Station Mo-del for Power System Analysis, in: IEEE Power EngineeringSociety General Meeting, 2006, pp. 1 – 5.

14



[15] C. Nicolet, F. Avellan, P. Allenbach, A. Sapin, J.-J. Simond,New tool for the simulation of transient phenomena in Francisturbine power plants, in: IAHR Symposium, Lausanne, Swit-zerland, 2002.

[16] C. Nicolet, Hydroacoustic modelling and numerical simulationof unsteady operation of hydroelectric systems, Ph.D. thesis,EPFL, Lausanne, Switzerland (2007).

[17] W. Zeng, J. Yang, R. Tang, W. Yang, Extreme water-hammerpressure during one-after-another load shedding in pumped-storage stations, Renewable Energy 99 (2016) 35 – 44.

[18] J. Yang, J. Yang, B-spline surface construction for the completecharacteristics of pump-turbine, in: 2012 Asia-Pacific Powerand Energy Engineering Conference, 2012, pp. 1–5.

[19] Y. Liu, J. Yang, J. Yang, C. Wang, W. Zeng, Transient Simula-tions in Hydropower Stations Based on a Novel Turbine Boun-dary, Mathematical Problems in Engineering 2016, Article ID1504659, 13 pages.

[20] C. Nicolet, E. F. R. Bollaert, P. Allenbach, A. Gal, Optimizinghydro production with a detailed simulation model, Hydropower& Dams 5 (2009) 77–81.

[21] P. Krause, O. Wasynczuk, S. Sudhoff, Analysis of Electric Ma-chinery and Drive Systems, 2nd Edition, IEEE Press, Piscata-way, New Jersey, USA, 2002.

[22] F. Bonnet, L. Lowinsky, M. Pietrzak-David, P.-E. Vidal, DoublyFed Induction Machine Speed Drive for Hydro-Electric PowerStation, in: European Conference on Power Electronics andApplications, Aalborg, Denmark, 2007, pp. 1–9.

[23] Y. Pannatier, B. Kawkabani, C. Nicolet, A. Schwery, J.-J. Si-mond, Start-up and Synchronization of a Variable Speed Pump-Turbine Unit in Pumping Mode, in: 19th International Confe-rence on Electrical Machines (ICEM), Rome, Italy, 2010, pp.1–6.

[24] A. Damdoum, I. Slama-Belkhodja, M. Pietrzak-David, M. Deb-bou, Low voltage ride-through strategies for doubly fed in-duction machine pumped storage system under grid faults, Re-newable Energy 95 (2016) 248 – 262.

[25] E. Levi, Impact of Iron Loss on Behavior of Vector VontrolledInduction Machines, IEEE Transactions on Industry Applicati-ons 31 (6) (1995) 1287–1296.

[26] S.-D. Wee, M.-H. Shin, D.-S. Hyun, Stator-Flux-Oriented Con-trol of Induction Motor Considering Iron Loss, IEEE Transacti-ons on Industrial Electronics 48 (3) (2001) 602–608.

[27] B. R. Oswald, Berechnung von Drehstromnetzen, Springer Vie-weg, Wiesbaden, Germany, 2013.

[28] S. Kouro, J. Rodriguez, B. Wu, S. Bernet, M. Perez, Poweringthe future of industry: High-power adjustable speed drive to-pologies, Industry Applications Magazine, IEEE 18 (4) (2012)26–39.

[29] W. Qiao, Dynamic Modeling and Control of Doubly Fed In-duction Generators Driven by Wind Turbines, in: Power Sys-tems Conference and Exposition, IEEE PES, Seattle, WA, USA,2009, pp. 1–8.

[30] S. Dieckerhoff, S. Bernet, D. Krug, Power Loss-Oriented Evalua-tion of High Voltage IGBTs and Multilevel Converters in Trans-formerless Traction Applications, IEEE Trans. Power Electro-nics 20 (6) (2005) 1328–1336.

[31] E. B. Wylie, V. L. Streeter, Fluid Transients in Systems,Prentice-Hall, Englewood Cliffs, New Jersey, USA, 1993.

[32] R. Knapp, Complete Characteristics of Centrifugal Pumps andTheir Use in the Prediction of Transient Behavior, Trans. ASME(1937) 683–689.

[33] N. Walmsley, The numerical representation of pump-turbineperformance characteristics, Ph.D. thesis, University of War-wick, UK (1986).

[34] L. Piegl, W. Tiller, The Nurbs Book, Springer Berlin Heidel-berg, New York, 1995.

[35] A. Adamkowski, Analysis of Transient Flow in Pipes With Ex-panding or Contracting Sections, Journal of fluids engineering125 (4) (2003) 716–722.

[36] T. Wu, C.-C. Ferng, Effect of nonuniform conduit section onwaterhammer, Acta Mechanica 137 (3-4) (1999) 137–149.

[37] W. Zielke, Frequency-Dependent Friction in Transient PipeFlow, Journal of Basic Engineering 90 (1) (1968) 109–115.

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